Thermal Flux in Unsteady Rayleigh-B\'{E} Nard

Home , Chandrasekhar number, Magnetic Prandtl number, Rayleigh number

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In flow. vertical vective homogeneous uniform a a in that convection showed thermal of problem [ geophysics like [ areas astrophysics traditional in relevance its to [ process in(B)[ unifor Rayleigh-Bénard magnetoconvec (RBM) a as tion tem known to adverse is subjected an field, to simultaneously magnetic subjected and is gradient fluid a perature of layer horizontal thin frequency satpco nes eerhdet t oeta industria potential its [ to nanofluids due in research intense plications of topic a is Introduction 1. di in flux thermal studied have We flu field. thermal magnetic on investigation vertical numerical of results present We Abstract n ag fCadaehrnme (50 number Chandrasekhar of range a and ei emaiiyo i.Temgei di as magnetic defined The is air. of permeability netic 4 h e the (4) κ viscosity kinematic tive nt stevleo iesols parameter dimensionless a of value the as unity σ Nu(Q) and , h yaiso B sgvre yfu dimensionless four by governed is RBM of dynamics The h nesadn fha u nmgeoovcieflows magnetoconvective in flux heat of understanding The steeetia odciiyo h udand fluid the of conductivity electrical the is ∗ mi address: Email author Corresponding i ff 5 erae oaihial ihCadaehrnme o Q for number Chandrasekhar with logarithmically decreases ciemgei rnt ubrPm number Prandtl magnetic ective n aeilpoesn eerh[ research processing material and ] f r antcneto,Ryeg ubr hnrska numbe Chandrasekhar number, Rayleigh Magnetoconvection, η = prxmtl as approximately 17 13 = h – [email protected] Nu(Q) – 1 20 16 / ( .Cadaehr[ Chandrasekhar ]. hra u nused Rayleigh-B unsteady in flux Thermal .Atemlysrtfidsse,weea where system, stratified thermally A ]. µ 0 σ i / ). ν h 1 Nu(0) n h e the and – a 3 eateto hsc,Ida nttt fTcnlg Khar Technology of Institute Indian Physics, of Department ,boud [ biofluids ], f i − ie hrl,rahsamxmmsihl bv nt n th and unity above slightly maximum a reaches sharply, rises 2 h rbblt itiuinfnto fteflcutn p fluctuating the of function distribution probability The . = ff ν/κ cietemldi thermal ective 17 sartoo h e the of ratio a is nlsdtelinear the analysed ] 4 ff ,electro-chemical ], svt ftefluid the of usivity rsn Kumar) Krishna ( ≤ 5 = Q – µ Transfer s 7 adpDas Sandip √ 0 σµ naddition in ] ≤ Ra stemag- the is 8 0 2 – / ν ff . QP)i asd h rbblt itiuinfnto ft of function distribution probability The raised. is Pr) (Q 12 where , 5 usivity × and ] ap- l ff ff ion nRyeg-ńr antcneto ntepeec of presence the in Rayleigh-Bénard magnetoconvection in x ec- 10 au h alihnme a h ieaeae usl num Nusselt averaged time the Ra, number Rayleigh the value d rn icu ud iharneo rnt ubr(0 number Prandtl of range a with fluids viscous erent a m -asinwt upa t maximum. its at cusp a with n-Gaussian rsn Kumar Krishna , - - 4 .Tepwrseta est fteNsetnme aiswi varies number Nusselt the of density spectral power The ). hc sartoo h uynyadLrnzfre.Frflu- For forces. Lorentz Pr and number buoyancy Prandtl the with of ids ratio a is which hwdsaigbhvorwt alihnme a[ Ra number Rayleigh with behaviour scaling showed a eue infiatyadtefli o a a was magnetoconvectio flow fluid turbulent the in and layer significantly fluid reduced was the across heat of ud hc sas nw sabase a as known also ordin is an which in fluid, ho nanoparticles of of metallic strength consists of nanofluid the suspension uniform mogeneous on A depend field. magnetic to applied found the was exponent scaling number The Nusselt averaged time the that found also was It oeo antcfil nteha u nused osi flu- in fluid flows metallic unsteady and crystals in the liquid flux on nanofluids, heat work including the ids any on field hardly magnetic is properti of the There role as well nanoparticles. as fluid d suspended base nanofluid of the of a properties of the on properties pend magnetic and electrical thermal, ovcini ealcflis[ fluids metallic in convection sartoo h oet oc e ntvlm oteda forc drag the to ber volume unit per e magneto-viscous force which to Lorentz due Q, the number of heat ratio Chandrasekhar’s global a of the is value of measure non-zero a for is computed flux which have Nu(Q), We dilution number low Nusselt nanoparticles. with spherical nanofluids verti- non-magnetic based uniform water of a in with RBM field in magnetic fluxes cal heat local and global both svre rm0 e from The varied is Q. of values smaller for lopotdtevraino h eue usl ubrNu number Nusselt reduced the have h We of Ra. variation of the value plotted fixed also a for Q with logarithmically creases rtclvleQ value critical a Nu(Q) xeiet ntemaueeto hra u nmagneto- in flux thermal of measurement the on Experiments nti ril epeetrslso ueia simulations numerical of results present we article this In > h Nu(Q) nr magnetoconvection enard ´ ,Nsetnme,lclha u,temlboundary thermal flux, heat local number, Nusselt r, Q i / c gu,Kaapr710,India Kharagpur-721302, agpur, hc eed nR n r h eue Nusselt reduced The Pr. and Ra on depends which , h Nu(0) i a, ∗ nrae lwywt hnrska ubrQ number Chandrasekhar with slowly increases i . 1 c ihadmninesparameter dimensionless a with hc eed nR n Pr, and Ra on depends which , ≤ Pr ff ≤ c.Tetm vrgdNsetnum- Nusselt averaged time The ect. r fteNsetnme snearly is number Nusselt the of art 6 nsatdcesn eysol to slowly very decreasing start en ≤ . .A ona srie above raised is Q as soon As 4. 21 ff 4 – . ciePadlnme ffluid of number Prandtl ective ,Nu 0, 23 hwdta h transport the that showed ] / are ud h viscous, The fluid. carrier r nrae hrl with sharply increases elclthermal local he . 1 uut2,2020 28, August h ff ≤ √ Nu(Q) ce [ ected Ra uniform a Pr 21 / < (QPr), s. , h i Nu 22 20 6 de- ber ary r . on 5) es th e- = ]. ]. n e i - the dimensionless parameter √Ra/(QPr). It attains a maximum Similarly the effective thermal diffusion coefficient κ = K/(ρcV ) slightly above unity and then begins decreasing slowly towards of a nanofluid with spherical non-magnetic metallic particles unity, as √Ra/(QPr) is further raised. The probability distri- may be computed using the expression: bution of fluctuations in the Nusselt number is close to nor- K f mal with slightly asymmetric tails. The probability distribution κ = [(1 φ)(ρcV) f + φ(ρcV )p] functions (PDF) of the local heat fluxes in the vertical direc- − (Kp + 2K f ) 2φ(K f Kp) tion are found to be non-Gaussian with a cusp at their maxima. − − . (8) PDFs are asymmetric about their maxima and have exponential × " (Kp + 2K f ) + φ(K f Kp) # − tails. Initially the fluid is at rest and the heat flux across the fluid layer is only due to conduction. The lower boundary of the nanofluid is maintained at temperature T , while the upper boundary is 2. Hydromagnetic System b maintained at temperature Tu = Tb ∆T. Here, β = (Tu − − Tb)/d = ∆T /d < 0. The steady state temperature profile Ts(z), We consider a thin horizontal layer of a homogeneous nano- density stratification ρs(z) and the pressure field Ps(z) across the fluid of effective density ρ and thickness d, effective thermal nanofluid in conduction state [17] are given by, expansion coefficient α and effective electrical conductivity σ and subjected to an adverse temperature gradient β in the pres- Ts(z) = Tb + βz, (9) = ence of a uniform magnetic field B0 B0e3 directed along the ρs(z) = ρ0 [1 + α (Tb Ts(z))] , (10) vertical direction. Here e3 is a unit vector in the vertically up- − 1 2 ward direction. The effective density ρ and the electrical con- Ps(z) = P0 ρ0g z + αβz , (11) − 2 ! ductivity σ are expressed [3] as: where Tb and ρ0 are the reference values of the temperature and ρ = (1 φ)ρ f + φρp, (1) density fields at the bottom surface of the nanofluid. P is a − 0 σ = (1 φ)σ f + φσp, (2) constant, which includes the magnetic pressure. As soon as β − is raised above a critical value βc, the basic state of conduction where φ is the volume fraction of the suspended spherically becomes unstable and convective flow (v , 0) begins. All the shaped nanoparticles of density ρp and electrical conductivity fields are perturbed and may be written as: σp in a base fluid of density ρ f and electrical conductivity σ f . ρs(z) ρ˜(x, y, z, t) = ρs(z) + δρ(x, y, z, t), (12) We may express the products ρα and ρcV for nanofluids [3] as: → Ts(z) T(x, y, z, t) = Ts(z) + θ(x, y, z, t), (13) (ρα) = (1 φ)(ρα) + φ(ρα) , (3) → f p Ps(z) P(x, y, z, t) = Ps(z) + p(x, y, z, t), (14) − → (ρcV ) = (1 φ)(ρcV) f + φ(ρcV)p, (4) B B(x, y, z, t) = B + b(x, y, z, t). (15) − 0 → 0 where cV stands for the effective specific heat of nanofluid at All length scales are measuredin units of the fluid thickness d constant volume. The effective thermal conductivity K of a and time is measuredin unitsof the free-fall time τ f = 1/ √αgβ, nanofluid [24] with spherical nanoparticles of thermal conduc- where g is the acceleration due to gravity. The fluid veloc- T tivity Kp in a base fluid of thermal conductivity K f is expressed ity v(x, y, z, t) = (v1, v2, v3) , the perturbation in pressure due as to flow p(x, y, x, t), the convective temperature θ(x, y, z, t) and the induced magnetic field b(x, y, z, t) are made dimensionless (Kp + 2K f ) 2φ(K f Kp) 2 2 K = K f − − . (5) by αgβd , ρ0αgβd , βd and νσµ0 B0, respectively. The value " (Kp + 2K f ) + φ(K f Kp) # − of thep effective magnetic Prandtl number Pm is of the order of 10 5 or less for terrestrial fluids including nanofluids. We Following Brinkman [25], the effective dynamic viscosity µ of − therefore set the value of Pm equal to zero in this work. This a nanofluid may be modelled as: makes the induced magnetic field b a slaved variable. The RBM 2.5 in nanofluids is then described by the following dimensionless µ = µ f (1 φ)− , (6) − equations: where µ f is the dynamic viscosity of the base fluid. All diffu- = + Pr 2 + QPr + sion coefficients may then be computedusing these expressions. Dtv p Ra v Ra ∂zb θe3, (16) −∇ q ∇ The effective kinematic viscosity ν or the effective momentum 2b = Ra ∂ v, ff ffi Pr z (17) di usion coe cient of the nanofluid may be computed as: ∇ − q 1 2 2.5 Dtθ = θ + v3, (18) µ µ f (1 φ)− RaPr ν = = − . (7) q ∇ ρ (1 φ)ρ f + φρp v = b = 0, (19) − ∇· ∇· 2 where Dt ∂t +(v ) is the material derivative. In the above the of spherical copper nanoparticles are varied from 0.2%to8.0%. dimensionless≡ number·∇ Rayleigh number Ra is defined as Ra = The set of hydromagnetic system (Eqs. 16-22) is applicable to 4 αβgd ρ0gαβd = 3 . It is a ratio of the buoyancy force per unit vol- water based homogeneousnanofluids with non-magneticmetal- νκ (ρ0νκ/d ) ume to the drag force per unit volume due to thermo-viscousef- lic particles. In the absence of nanoparticles (φ = 0), the hydro- fect. Other dimensionless external parameter for magnetocon- dynamic system represents magnetoconvection in geophysical vection is the Chandrasekhar’snumberQ, which is a measureof fluids. The value of Pr for Earth’s liquid outer core is approxi- the strength of the external magnetic field and it is a ratio of the mated to be in a range from 0.1to10[8]. Some liquid crystals Lorentz force per unit volume to the drag force due to magneto- have Pr 4.0. These equations may also be useful in elec- 2 2 2 ∼ σB0d B0/(µ0d) trically conducting gases. The gases at high temperatures may viscous effect. It is defined as Q = = 3 . It is also ρ0ν (ρ0νη/d ) conduct electricity as in a discharge tube. The range of Pr is equal to square of the Hartmann number H = B0d σ/(ρ0κ). It chosen to cover different types of fluids. The critical Rayleigh plays the role which Taylor number plays in RBC with Coriolis p number Rac(Q) for the onset of stationary magnetoconvection force [17]. depends on the Chandrasekhar’s number Q. The critical wave Horizontal boundaries, located at z = 0 and z = 1, are consid- number kc(Q), which is the wave number at the onset of con- ered to be thermally conductingand electrically nonconducting. vection, also depends on Q. The expressions for Rac(Q) and Teflon or ethylene-vinyl-acetate (EVA) composites may realize kc(Q) are: these conditions in an experiment [26]. Horizontal boundaries 2 2 π +kc 2 2 2 2 made of good thermal conducting material and maintained at Rac(Q) = 2 (π + k ) + π Q , (24) kc c constant temperatures do not allow temperature fluctuations at h i 1 the boundaries due to convective flow in the fluid. So the con- kc(Q) = π a+ + a , (25) − − 2 vective temperature field θ vanishes at the boundaries. Elec- q 1 3 trically nonconducting surfaces do not allow current across the 2 1 a = 1 1 + Q 1 + Q 1 2 , (26) surface. Therefore, the vertical component of the current den-  4 2 π2 2 π2 4  ± ± h − i sity j = (∇ × b)/µ should also vanish at the horizontal bound-   0   aries. In addition, the induced magnetic field should be con- The global heat flux across the fluid layer is defined by Nus- tinuous at the boundaries. The induced magnetic field bp in selt number Nu, which is a ratio of spatially averaged the total an electrically non-conducting plate of permeability µp must be heat flux and the conductive heat flux. It is defined in terms of derivable from a potential [17]. That is, dimensionless vertical velocity v3 and convective temperature θ as: b = bp at z = 0, 1, where (20) 2π 2π 1 p 2 √Ra Pr kc kc b = ∇Ψ, where Ψ= 0. (21) = + ∇ Nu(t) 1 v3θ dxdydz, (27) V Z0 Z0 Z0 In the limit Pm 0, as considered here, the boundary con- → 2 2 ditions of the induced magnetic field b are dictated by Eq. 17. where V = 4π /kc is the dimensionless volume of the simu- This equation is satisfied when b1, b2 and ∂zb3 vanish at the lation box. The Nusselt number, which is a function of time horizontal boundaries. This choice also ensures that j3 = 0 and for unsteady magnetoconvection, depends on Ra, Pr and Q. ∇·b = 0 are automatically satisfied. The velocity boundarycon- Its time averaged value over a long period T is denoted as T ditions on horizontal boundaries are assumed to be stress-free, Nu = 1 Nu(t)dt. The quantity v θ represents the local h i T 0 3 which are idealized boundary conditions. A good approxima- heat flux inR the vertical direction due to magnetoconvection. tion for stress-free boundary conditions were realized in experiments by Goldstein and Graham [27]. RBM at higher values of 3. Direct Numerical Simulations Chandrasekhar’s number Q flows are not affected significantly due to stress-free boundary conditions. The relevant boundary Direct numerical simulations are done using pseudo-spectral conditions [17] are then given as: method. All fields are assumed to be periodic in the horizontal

∂zv1 = ∂zv2 = v3 = b1 = b2 = ∂zb3 = θ = 0 at z = 0, 1.(22) plane. The expansion of the relevant perturbations, consistent with the boundary conditions considered, are: Let us denote magnetic field in the upper boundary as b z>1 | ik(lx+my) and the same in the lower boundary as b z<0, respectively. Then v (x, y, z, t) = U (t)e cos(nπz), (28) | 1 lmn lX,m,n b z 1 = ∇Ψ z 1 and b z 0 = ∇Ψ z 0, (23) | ≥ | ≥ | ≤ | ≤ ik(lx+my) v2(x, y, z, t) = Vlmn(t)e cos(nπz), (29) where Ψ z 1 and Ψ z 0 are scalar potentials in the regions z > 1 | ≥ | ≤ lX,m,n and z < 0, respectively. The non-zero horizontal velocities of a ik(lx+my) nanofluid at the stress-free boundaries allow surface currents at v3(x, y, z, t) = Wlmn(t)e sin (nπz), (30) the horizontal boundaries. The continuity of the vertical com- lX,m,n ik(lx+my) ponent of the induced magnetic field at the horizontal bound- θ(x, y, z, t) = Θlmn(t)e sin (nπz), (31) aries (z = 0, 1) fixes the horizontal current. lX,m,n The effective thermal Prandtl number of the water based ik(lx+my) p(x, y, z, t) = Plmn(t)e cos(nπz), (32) nanofluids may be varied from 6.5to4.0, if the volume fraction lX,m,n 3 u u θ θ Pr Ra Q Nu ǫ (est.) ǫ (comp.) ǫ (est.) ǫ (comp.) Nukin Nuth

1.0 3.04 106 0 19.76 0.0107 0.0107 0.0113 0.0112 19.66 19.56 × 300 20.44 0.0111 0.0110 0.0117 0.0115 20.18 20.10

700 20.59 0.0112 0.0111 0.0118 0.0116 20.35 20.23

4.0 5.0 105 0 13.53 0.0088 0.0088 0.0095 0.0095 13.45 13.40 × 100 13.63 0.0089 0.0089 0.0096 0.0094 13.58 13.30

300 11.80 0.0076 0.0075 0.0083 0.0081 11.61 11.46

500 10.85 0.0069 0.0068 0.0076 0.0074 10.62 10.46

6.4 5.0 105 0 14.31 0.0074 0.0074 0.0079 0.0079 14.24 14.17 × 100 13.41 0.0069 0.0068 0.0074 0.0073 13.16 13.06

300 11.43 0.0058 0.0057 0.0063 0.0062 11.20 11.09

500 10.78 0.0054 0.0053 0.0060 0.0058 10.48 10.37

Table 1: List of Nusselt number Nu(Q) for diﬀerent values of Chandrasekhar’s number Q, kinetic energy dissipation rate ǫu and ‘thermal energy’ dissipation rate ǫθ. These dissipation rates are compared with their estimated values using the formulas ǫu = (Nu 1)/ √RaPr and ǫθ = Nu/ √RaPr. −

where Ulmn(t), Vlmn(t), Wlmn(t), Θlmn(t), and Plmn(t) are the more at lower values of Q, if the value of Ra is sufficiently Fourier amplitudes in the expansion of the fields v1, v2, v3, θ, raised. As a results the spatial resolution required is less, if Q is and p respectively. The wave vector of perturbations in the hor- raised to a highervalue with Ra andPr fixed. Thespatial resolu- izontal plane is k = lke1 + mke2. We have set k = kc(Q), where tions used here are sufficient to describe the magnetoconvective kc(Q) is the critical wave number for a given value of Q. The flow for the parameter values considered. We have compared integers l, m, n can take values compatible with continuity equa- our results for Q = 0 with those obtained by Veronis [28], tion, which leads to the following condition. Moore & Weiss [29] and Thual [30], who used the identical boundary conditions. The two sets of grid resolutions used here ilk (Q)U + imk (Q)V + nπW = 0. (33) c lmn c lmn lmn keep the minimum value of the global Kolmogorov dissipative The expansions of the magnetic fields in the boundaries outside scale always more than 2. We have also reproduced the re- the nanofluids [20] may be expressed as: sults reported in the earlier works on magnetoconvective insta- bility [20] as well as on RBC [31]. Of course, for much lower n+1 ik(lx+my) γ(1 z) Ψ z 1 = ( 1) Ψlmn(t)e e − , (34) | ≥ − valuesofPr (< 0.1) andmuch highervaluesof Ra wouldrequire lX,m,n better spatial grid resolutions. We have listed in Table 1 values ik(lx+my) γz Ψ z 0 = Ψlmn(t)e e , (35) of Nusselt number Nu(Q), the average dissipative rates for the | ≤ 2π 2π 1 lX,m,n ǫu = ν kc kc ∂ v x, y, z, t 2dxdydz kinetic energy 0 0 0 i j( ) and nπW (t) h 2π 2π {1 } i 2 2 lmn θ R kRc kRc 2 where γ = k (l + m ) and Ψlmn(t) = 2 2 2 . The spa- ǫ = κ ∂ θ x, y, z, t dxdydz γ(γ +n π ) ‘thermal energy’ 0 0 0 i ( ) for tial grid resolutionsp of the periodic box of size L L 1, different values of Chandrasekhar’sh number{ Q. The} computedi × × R R R where L = 2π/kc(Q). Spatial resolution of 128 128 128 or values of the dissipation rates ǫu and ǫθ are compared with their × × 256 256 256 grid points has been used for simulations pre- estimated values using the formulas: ǫu = (Nu(Q) 1)/ √RaPr × × sented here. As the Rayleigh number is raised above a critical and ǫθ = Nu(Q)/ √RaPr in Table 1. They are in− good agree- value Rac(Q), while keeping the values of Q and Pr fixed, sta- ment. Rayleigh-Bénard convection was investigated numeri- tionary magnetoconvection begins [17]. We define the distance cally in a cubic box with no-slip velocity boundary conditions Ra Ra (Q) from criticality by a parameter ǫ = − c . As Q is raised Rac(Q) on all walls by Xu etal [32]. Their velocity boundary condi- keeping Ra and Pr fixed, the parameter ǫ becomes smaller and tions and simulation box size were different than what we have consequently the fluctuations are reduced. The fluctuations are 4 considered here. The values for the Nusselt number in a fluid with Pr = 7.0 are 8.49 and 11.12 for Ra = 106 and 3 106, respectively. Our values for Nusselt number at are almost× double (17.32 for Ra = 106 and 23.54 for Ra = 3 106). The defini- tions of the dissipation rate for kinetic energy× also differs by a numerical factor of 1/2. We record the values of all relevant fields at all spatial grid points at an regular interval of every two units of dimensionless time. We have computed minimum number of 300 frames for each set of parameter values reported here.

20 Q = 50 15

10 0 100 200 300 400 20 Q = 70 15

10 0 100 200 300 400 16 Q = 300 12

Nu(Q) Figure 2: (Color online) Three-dimensional temperature isosurfaces computed 8 from DNS for Ra = 5.0 105 and Pr = 4.0 for (a) Q = 70 and (b) Q = 500. Red 0 100 200 300 400 × 16 (grey) and blue (black) colors stand for hotter and cooler ﬂuids, respectively. Q = 500 a 12

8 0 100 200 300 400 Dimensionless time(t)

Figure 1: (Color online) Variation of Nusselt number Nu(Q) with dimensionless time t [blue (black) curves] for Rayleigh number Ra = 5.0 105 and Prandtl number Pr = 4.0 for diﬀerent values of Chandrasekhar number× Q.

4. Results and Discussions 2

As Rayleigh number Ra is raised to a suﬃciently high value 1.6 for ﬁxed values of Chandrasekhar number Q and Prandtl number Pr, the magnetoconvection becomes unsteady. Fig 1 shows × 6 1.2 Ra=3.04 10 , Pr=1.0

(Nu) Ra=3.04×106, Pr=0.8

the variation of Nusselt number for chaotic magnetoconvec- σ tive flow with dimensionless time for Ra = 5.0 105 and Ra=8.0×105, Pr=1.0 5 Pr = 4.0 for different values of Q. The temporal evolution× of 0.8 Ra=5.0×10 , Pr=4.0 × 5 Nusselt number for Q = 70, 300, 500 show that the time aver- Ra=5.0 10 , Pr=1.0 × 5 0.4 Ra=5.0 10 , Pr=0.8 aged mean value Nu(Q) decreases with increase in Q. It con- Ra=2.5×105, Pr=1.0 firms that the magnetic field suppresses the heat flux of in RBM Ra=7.0×104, Pr=0.1 0 at relatively larger values of Q. Figure 2 shows typical three- 0 200 400 600 800 1000 1200 1400 1600 dimensional isosurfaces computed at a given instant from the Q DNS for Ra = 5 105 and Pr = 4.0 for two different values × Figure 3: (Color online) Standard deviation σ(Nu) of the Nusselt number Nu of Q. More thermal plumes are generated for Q = 70 than for for different values of Ra, Pr and Q. Q = 500. The generation of more thermal plumes leads to enhancement of the relative Nusselt number in a range of lower values of Q for fixed values of Ra and Pr. The temporal fluctuation of Nusselt number is considerable for lower valuesof Q (see Fig. 1). Figure 3 displays the standard 5 0 10

0 10 Q = 50 −1 10 Q = 70 Q = 300 Q = 500

−2 −5 10 Q=50, Ra=5.0×105 10 PDF PSD Q=70, Ra=5.0×105 Q=300, Ra=5.0×105 −3 −1.93 10 Q=500, Ra=5.0×105 f 9 Ra=1.7×10 (expt. data) −10 10 Ra = 5.7×107(expt. data) −4 10 −4 −2 0 2 4 6 −2 −1 0 1 2 3 〈 〉 σ 10 10 10 10 10 10 [Nu− Nu ]/ (Nu) f

Figure 4: (Color online) Probability density function (PDF) of the fluctuations in Nusselt number Nu at Ra = 5.0 105 and Pr = 4.0 for different values Figure 5: (Color online) Frequency power spectral density (PSD) of the Nusselt × number (Nu) for Ra = 5.0 105 and Pr = 4.0 at different values of Q. of Q. Red (gray) squares, blue (black) circles, magenta (dark gray) triangles × and green (gray) diamonds are for Q = 50, 70, 300 and 500, respectively. Blue (black) stars and cyan (light gray) squares are experimental data points (Ref. [33]) in absence of the external magnetic field (Q = 0) for Ra = 5.7 107 × and 1.7 109, respectively. 2 × PSD of Nu scales with frequency f approximately as f − . The best fit to curves obtained from simulations gives the value of the exponent as 1.93 0.02. The value of the exponentis quite close to one observed− ± in experiments on turbulent RBC [33] as deviation in the temporal signals of Nusselt number for differ- well as in numerical simulations of turbulent RBC with rota- ent values of Ra, Pr and Q. Fluctuations are larger at higher tion [34]. values Ra. Fluids with lower values of effective thermal Prandtl The upper viewgraph in Fig. 6 displays the variation of time number Pr show relatively larger fluctuations. The fluctuations averaged Nusselt number Nu(Q) with Chandrasekhar number are suppressed at higher values of the applied magnetic field, if Q for several values of Rah and Pri on a semi-log scale. The ma- all other parameters are kept fixed. For higher values of Q, the genta (very light gray), green (light gray), blue (black), brown distance from the onset of magnetoconvection is smaller and, (dark gray) and violet (gray) curves are the best fit to the data consequently, the fluctuating part of the Nusselt number is re- points obtained from simulations for different curves. Cyan duced. (light gray) diamonds, blue (black) circles and red (gray) tri- Figure 4 shows the probability distribution function (PDF) angles at the top are the data points computed for Pr = 2.0, 1.0 of the fluctuating parts of the Nusselt number around the mean and 0.8, respectively. The green (gray) inverted triangles repre- for different values of Q. Probability distribution function is sent the data points computed for Pr = 1.0 in the second data close to normal distribution but with slightly asymmetric tails. set from top (Ra = 8.0 105). Brown (dark gray) stars, violet For effective value of Pr = 4, the height of PDF is the low- (gray) left pointing triangles,× magenta (light gray) squares and est for Q = 70 but tails are longer. As Q is increased or de- green (gray) stars in the third data set (Ra = 5.0 105) are com- creased, the height of PDF goes up and tails become shorter. puted for Pr = 0.8, 1.0, 4.0 and 6.4, respectively.× Data points, We have also compared the computed PDFs with the experi- shown as azure (gray) right pointing triangles in the fourth data mental results of Aumaitre and Fauve [33] on Rayleigh-Bénard set from the top (Ra = 2.5 105), are for Pr = 1.0. Data points, convection (RBC) in water and mercury in the absence of exter- shown as blue (black) diamonds× and black (black) squares in nal magnetic field. The data points shown by navy blue (black) the data set at the bottom are for Pr = 0.1 and Pr = 0.2, re- stars and cyan (light gray) squares are adopted from their exper- spectively. The lower viewgraph in Figure 6 shows the plot of iments at much higher values of Ra. PDFs obtained for RBM threshold Rac(Q) for stationary magnetoconvection with Q, as are in good agreement with those observed in RBC in water and obtained by Chandrasekhar [17] for stress-free velocity bound- mercury. The slight difference is due to small Rayleigh number ary conditions. The time averaged value of the Nusselt num- used for simulations. The computed curves are smoother due to ber Nu(Q , for fixed values of Ra and Pr, first increases very large number of data points. slowlyh withi Q, reaches a maximum and then starts decreas- Figure 5 shows the power spectrum density (PSD) in the fre- ing quickly with Q. The tendency of slight enhancement of quency space of the Nusselt number for Ra = 5.0 105 and heat flux was not observed for Pr = 6.4. It has some simi- Pr = 4.0 and for different values of Q. The PSD shows× noisy larity with enhancement of thermal flux at low rotation rates behaviour at lower frequencies. However, the Nusselt number in rotating RBC. Fig. 6 also shows that decrease of Nu(Q) is found to show scaling behaviour at higher frequencies. The with Q is logarithmic for higher values of Q [see the magentah i 6 25 Ra=3.04×106 Ra=8.0×105 20 Ra=5.0×105 Ra=2.5×105 15 Ra=7.0×104 〉 Nu 〈 10

0 1 2 3 4 5 10 10 10 10 10 8 Q 10

(Q) 5

c 10 2 Ra 10 2 3 4 5 10 10 10 10 Q

Figure 6: (Color online) The variation of time averaged Nusselt number Nu with Chandrasekhar’s number Q for diﬀerent values of Ra and Pr is shown in the upper viewgraph. The data set at the top is for Ra = 3.04 106. Data pointsh showni by cyan (light gray) diamonds, blue (black) circles and red (dark gray) triangles are computed for Pr = 2.0, 1.0 and 0.8, respectively. The× second set of computed data points from the top [green (gray) inverted triangles] are for Ra = 8.0 105 and Pr = 1.0. The third data set from the top is for Ra = 5.0 105. Here, brown (dark gray) stars, violet (gray) left pointing triangles, magenta (light gray) squares× and green (gray) stars are data points computed for Pr = 0.8,× 1.0, 4.0 and 6.4 respectively.The fourth data set from the top [azure (gray) right pointing triangles] is for Ra = 2.5 105 and Pr = 1.0. The set of data points at the bottom are for Ra = 7.0 104. Here blue (black) diamonds and black (black) squares are data points × × computed for Pr = 0.1 and Pr = 0.2, respectively. The lower viewgraph shows the plot of the threshold Rac(Q) with Q.

7 1.1

Ra = 3.04 × 106 Pr = 0.8 × 6 〉 0.9 Ra = 3.04 10 Pr = 1.0 Ra = 3.04 × 106 Pr = 2.0 Ra = 8.0 × 105 Pr = 1.0

Nu(0) 0.8 Ra = 5.0 × 105 Pr = 0.8 〈 5

/ Ra = 5.0 × 10 Pr = 1.0 〉 × 5 0.7 Ra = 5.0 10 Pr = 4.0 Ra = 5.0 × 105 Pr = 6.4 Ra = 2.5 × 105 Pr = 1.0 Nu(Q) 〈 0.6 Ra = 7.0 × 104 Pr = 0.1

= × 4

r Ra = 7.0 10 Pr = 0.2 1.06 Nu 0.5 1.04 1.02 0.4 1 0.98 0.96 20 40 60 80 100 120

0 20 40 60 80 100 120 [Ra/Q Pr)]1/2

Figure 7: (Color online) Variation of the reduced Nusselt number Nur = Nu(Q) / Nu(0) with the dimensionless quantity √Ra/(QPr) for several sets of Ra and Pr: (i) Pr = 0.1 with Ra = 7.0 104 [blue (black) diamonds], (ii) Pr = 0h.2 withi Rah = 7.0i 104 [black (black) squares], (iii) Pr = 0.8 and Ra = 3.04 106 [red (gray) triangles] and 5.0 105 [brown× (dark gray) stars], (iv) Pr = 1.0 with Ra = 3.04 106×[blue (black) circles], 8.0 105 [green (light gray) inverted× triangles], 5.0 105 [violet (gray) left× pointing triangles] and 2.5 105 [azure (gray) right pointing× triangles], (v) Pr = 2.0 with Ra×= 3.04 106 [cyan (light gray) diamonds], (vi)× Pr = 4.0 with Ra = 5.0 105 [magenta (light gray)× squares] and (vii) Pr = 6.4 with Ra = 5.0 105 [green (gray) stars]. Inset× shows the enlarged view of the × × plot for Nur 1.0. ≈

8 0.15 Ra=2.5×105 1 Ra=5.0×105 Ra=8.0×105 0.1 0.8 Ra=3.04×106 Q = 300 th 0.4 δ 0.6 0.05 0.3 z

z 0.2 0.4 0 0.1 200 600 1000 Q

0 −1 0.2 0.4 0.6 0.8 10 Q=300 〈T〉(z) Q=500 th

δ Q=700

0 Q=1000 0 0.2 0.4 0.6 0.8 1 −2 〈T〉(z) 10 5 6 7 10 10 10 Ra Figure 8: (Color online) Variations of horizontally averaged temperature field T (z) fields with the vertical coordinate are plotted for Q = 300 [blue (black) Figure 9: (Color online) Thermal boundary layer: Plot of the thickness of ther- circle],h i for Pr = 1.0 and Ra = 3.04 106. Red (gray) straight lines show the mal boundary layer (δth) computed for Pr = 1.0 and Ra and Q. The upper variations of temperature in the central× part and near the lower boundary. viewgraph shows the variation of δth with Q for different values of Ra. The lower viewgraph shows variation of δth with Ra (on log-log scale) for different values of Q.

(light gray), green (gray), blue (black) and brown (dark gray) lines]. For given values of Q and Pr, the mean Nusselt num- 3 ber is higher for larger values of Ra. The effect of Pr is clearly 10 visible only for Q < Q (where Q = Q denotes the critical 1.8 c c 10 value of Chandrasekhar number, above which the logarithmic behaviour starts to set in), if Ra is kept fixed. For Q < Qc, Nu h i increases with Pr. There is no significant change in Nu for 0 h i 10

Q > Qc with Pr, if Ra is kept fixed. However, Nuc (the critical ) 1.4 θ 10 value of time averaged Nusselt number at Qc) increases but Qc 3 −0.01 0 0.01 decreases with Pr for a fixed value of Ra. The valuesof Nuc and = PDF(v Q = 50 Qc both increases with Ra. For Pr 1.0, we observe that Nuc −3 0.28 0.01 0.65 0.03 10 and Qc vary with Ra as Ra ± and Ra ± , respectively. Q = 70 For Q > Qc, the Lorentz force starts playing dominant role on Q = 300 the heatflux across the fluid layer. On the other hand, the role Q = 500 of Lorentz force is less significant for Q < Qc. −6 The slope of the Nu(Q) Q curves for Q depends mainly 10 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 h i− v θ on Ra. The time averaged Nusselt number for Q > Qc may 3 therefore be expressed as:

Nu(Q) = C (Ra, Pr) C (Ra, Pr) ln Q, (36) Figure 10: (Color online) The probability distribution functions (PDFs) of ver- h i 1 − 2 tical heat flux for Pr = 4.0 computed for Q = 50 [pink (light gray) circles], 70 [blue (black) squares], 300 [magenta (gray) stars], and 500 [green (gray) where C1 and C2 are functions of Ra and Pr. triangles] with Ra = 5.0 105. Inset shows the PDFs near their maxima. We now define a reduced (or normalized) Nusselt number × Nu = Nu(Q) / Nu(0) as a ratio of the Nusselt number in r h i h i the presence of an external magnetic field (Q , 0) and the of Pr. Nur increases sharply from a small value ( 1) and at- Nusselt number Nu(0) in the absence of any external magnetic tains a value slightly bigger than unity, as √Ra/(QPr)≪ is raised field (Q = 0). The dimensionless parameter Ra/(QPr) is a ra- slowly. With further increase in √Ra/(QPr), the value of Nur 4 tio of the buoyancy force per unit volume (αβgd ρ0) and the starts decreasing slowly and tends to approach unity (see the 2 2 Lorentz force per unit volume (σB0d ν). If the vertical mag- plots for Pr = 0.8, 1.0, 4.0) slowly. The inset in Fig. 7 shows netic field always suppressed the transport of heat across the an enlarged view of the curve showing Nur more than unity. fluid layer [17, 21–23], the value of Nur should always be less The maximum enhancement of thermal flux is observed for than unity and it should approach asymptotically to unity as the 0.1 Pr 4.0. The value of the parameter √Ra/(QPr), where ≤ ≤ parameter √Ra/(QPr) is raised to a much larger value. Fig. 7 Nur reaches its maximum, depends on Ra and Pr. We do not shows the variation of Nur with √Ra/(QPr) for different values observe enhancement of thermal flux for Pr = 6.4. For lower 9 values of Q and for a range of Pr, the enhancement of ther- direction. All the PDFs are asymmetric about their maxima lo- mal flux is observed in the unsteady magnetoconvection. This cated at v3θ = 0 and are non-Gaussian. The asymmetry of the behaviour has similarity with enhancement of thermal flux ob- PDF shows that the excursion of upward heat flux is more than served in rotating RBC at lower values Rossby number (higher the excursion of downward heat flux. This signifies that a net rotation rates) [35–40]. However, the amount of enhancement heat flux is maintained in the vertically upward direction. The observed in the case of magnetoconvection is less compared to data points in PDFs shown by blue (black) squares, pink (light that observed in rotating RBC. In addition, the enhancement gray) circles, magenta (gray) stars and green (gray) triangles of thermal flux in magnetoconvection is not observed at larger are for Q = 70, 50, 300 and 500, respectively. The time aver- values of Pr in RBM. This may be due to efficient generation of aged PDFs of local thermal fluxes in the vertical direction show thermal plumes in rotating RBC at relatively higher values of a cusp at the maximum. This type of cusp was first observed in Pr [39] in rotating RBC. experiment on turbulent RBC [43]. The PDF of instantaneous Thin boundary layers are also characteristics of a turbulent local fluxes in the vertical direction also showed the cusp at the flow [41, 42]. The thickness of thermal boundary layer δth in maxima in simulations [44]. It may be due to non-Gaussian na- γ turbulent RBC is known to scale with Ra as δth Ra− . The ture of the vertical velocity v3 and the convective temperature exponent γ is found to lie between 0.2 and 0.3 [42∼ ]. We also θ. The inset of Fig. 10 shows an enlarged view of the PDFs investigated the role of magnetic field on the thickness of ther- near their maxima. The time averaged PDFs with wider spread mal boundary layer. To compute the thickness of the boundary have lower values of maxima. For Ra = 5.0 105 and Pr = 4.0, × layer (δth), we first spatially averaged the total temperature field the largest spread of a PDF is for Q = 70. The histograms for T(x, y, z, t) in horizontal plane for each frame of computed data these cases have exactly the similar shapes (not shown here) points. This led to a temperature field, which is a function of and they show the time averaged vertical local heat flux is max- the vertical coordinate z and dimensionless time t. A time aver- imum for Q = 70, which correspond to √Ra/(QPr) = 42.25 age of a large number of frames (300 frames or more) yielded for Pr = 4.0. This is consistent with the largest global heat flux a temperature field T (z), which depends only on the vertical for Q = 70 for the same set of all parameters (see Fig. 6). The coordinate z. One suchh i case for Ra = 3.04 106, Pr = 1.0 and probability distribution functions of the local heat fluxes show Q = 300 is shown in Fig. 8. It clearly shows× a sharp drop in the exponential tails. A large part of the distribution functionforthe temperaturefield in a thin layer of the fluid near both the bound- upward local heat flux may be represented with two exponen- aries. The temperature drop in the central part of the simula- tial functions, while the distribution function for the downward tion cell is very small. We draw two straight lines: one drawn local heat flux can be represented well by a single exponential through the almost vertical part and another drawn through the function. It is interesting to note that local energy flux shows part where the temperature drop is sharp (see Fig. 8). The es- approximately exponential tails in wave turbulence [45]. How- timated thermal boundary layer δth is the vertical distance of ever, the anisotropy in a thermally stratified system makes the the point of intersection from the nearest boundary. The upper probability distributions of local heat fluxes in vertical direction viewgraph of Fig. 9 shows the variation of the thickness of ther- asymmetric on two sides of the peak. mal boundary layer with Q for Pr = 1.0 and different values of Ra. The thickness δth increaseswith Q for a fixedvalue of Ra. It 5. Conclusions is expected as the increase in Q brings down the distance from criticality ǫ. The lower viewgraph shows the variation of δth A numerical study on global as well as local heat fluxes with Ra for different values of Q on log-log scale. The bound- in Rayleigh-Bénard magnetoconvection in different fluids is ary layer thickness decreases with increase in Ra for a fixed presented. The global heat flux of unsteady magnetocon- value of Q. The boundary layer thickness shows scaling behav- vection with a uniform vertical magnetic field shows a mild γ ior with Ra: δth Ra− , where the exponent γ(Q) now depends enhancement as the strength of the uniform magnetic field is on the Chandrasekhar’s∼ number Q. The value of γ is found to raised for relatively lower values of Q and a range of thermal be 0.30 0.01 for Q = 3 102 and 0.18 0.01 for Q = 103. Prandtl number (0.1 Pr 4.0). For relatively higher values The value± of γ is in excellent× agreement with± the experimental of external magnetic≤ field,≤ there is suppression of heat flux in observation of Zhou and Xia [42] for lower value of Q. nanofluids, liquid crystals as well in geophysical liquid metals. We have also computed the probability distribution functions The time averaged global heat flux (Nusselt number) decreases (PDFs) of the local heat fluxes in the vertical direction to inves- logarithmically with Chandrasekhar’s number for all fluids tigate the role of the external magnetic field on PDFs. For this, responsive to a vertical magnetic field, when Rayleigh number the values of the vertical velocity v3 and convective tempera- and Prandtl numbers are kept at fixed values. For water based ture θ are recorded at all spatial grid points at regular interval nanofluids is likely to show this behaviour, if the volume frac- for a long time. A probability distribution function (PDF) of tion of spherical copper nanoparticles is around 8%. A similar v3θ is then computed for each of these frames. A time averaged behaviour is likely in Earth’s outer liquid core (Pr 0.1) as PDF of local heat fluxes is then obtained using a minimum of well as some liquid crystals. 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